Saxena, Subhash Chandra, 1934-||Beach, James W.
M.S. (Master of Science)
Department of Mathematics
In the study of the separation axioms and their related theorems it is found that the various results obtain- ed depend upon either the particular separation axiom satisfied or the individual topological space involved. Knowledge of elementary set theory and a previous introduction to metric spaces and their related concepts are required of the student before complete comprehension of this study can be attained. A topological space is defined as well as the various topological concepts employed in the paper. The separation axioms are stated and a few basic theorems are proven. Some elementary theorems are also proven to illustrate that certain separation axioms are satisfied by particular topological spaces. Topological spaces in general are classified according to which separation axiom Is satisfied. Regular and normal spaces are defined and a few related theorems are proven. The implications made by specific T? spaces are studied to some extent. However, the proofs of the theorems Involved vary widely in difficulty. Completely normal spaces are defined and a few associated theorems are proven. Finally, compactness in the T? spaces is studied. It is soon found that theorems on compactness deal almost entirely with T? spaces. Locally compact topological spaces are defined and an important theorem on the one-point compactification of a topological space is proven. The definition of completely regular spaces and two related theorems conclude the study. It is pointed out that only a mathematically mature person would receive the full benefit of such a study of the separation axioms. Reference to specific areas that are not Incorporated in this study, such as paracompact topological spaces and Hausdorff uniform spaces, is given to guide the Interested student in further study.
Allen, Merle Clinton, "The separation axioms and related theorems" (1965). Graduate Research Theses & Dissertations. 6289.
Northern Illinois University
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